Ordinary varieties and the comparison between multiplier ideals and test ideals. II.
In this paper, the author relates the following two conjectures. For simplicity of notation in this review, we consider our varieties with defining equations in \(\mathbb{Z}\). The paper works in greater generality. In particular, the first conjecture is as follows.NEWLINENEWLINEConjecture 1.1. For every ideal \(a\) on a smooth variety \(X\) of characteristic \(0\), there exists a Zariski-dense set of prime integers such that if \(X_p\) (respectively \(a_p\)) is the reduction mod \(p\) of \(X\) (respectively \(a\)) then \(\mathcal{J}(X, a^t)_p = \tau(X_p, a_p^t)\) for all \(t \geq 0\). Here \(\mathcal{J}(X, a^t)\) is the multiplier ideal and \(\tau(X_p, a_p^t)\) is the test ideal.NEWLINENEWLINEThe second conjecture states:NEWLINENEWLINEConjecture 2. For every smooth projective \(d\)-dimensional variety \(X\) of characteristic \(0\), there is a dense set of primes such that the Frobenius induces isomorphisms \(H^{d}(X_p, \mathcal O_{X_p}) \to H^{d}(X_p, \mathcal O_{X_p})\).NEWLINENEWLINEThis bijectivity of Frobenius is weaker than the assertion that \(X_p\) is ordinary in the sense of \textit{S. Bloch} and \textit{K. Kato} [Publ. Math., Inst. Hautes Étud. Sci. 63, 107--152 (1986; Zbl 0613.14017)].NEWLINENEWLINEIn [Nagoya Math. J. 204, 125--157 (2011; Zbl 1239.14011)], the author and \textit{V. Srinivas} related two these conjectures. They showed that conjecture 1.2 implies conjecture 1.1. In this short paper, the author proves that conjecture 1.1 implies conjecture 1.2 showing that the two conjectures are equivalent.
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